Convert Recurring Decimals to Fractions
🎬 Math Angel Video: Convert Recurring Decimals to Fractions
How to Convert Recurring Decimals to Fractions? (0:01)
Recurring decimals are decimals that go on forever with a repeating pattern.
For example:
$$0.3333\ldots$$
has the digit 3 repeating forever.
We can turn recurring decimals into fractions. Here’s how:
🔮 Example: Writing 0.333… in a fraction:
Let $x = 0.\dot{3}$ (the recurring decimal).
Multiply by 10: $$10x = 3.\dot{3}$$
Subtract the two equations. We do this to remove the recurring part:
$$
\begin{aligned}
10x – x &= 3.\dot{3} – 0.\dot{3} \\
9x &= 3
\end{aligned}
$$Solve for $x$:
$$x = \tfrac{1}{3}$$
✅ So, $0.3333\ldots = \tfrac{1}{3}$.
Converting 0.4545... into a Fraction Step by Step (0:46)
Let’s see a more advanced recurring decimal.
$$0.454545\ldots$$
has the digits 45 repeating forever.
We can use the same method and steps to turn this recurring decimal into a fraction.
🔮 Example: Writing 0.4545… in a fraction:
Let $x = 0.\dot{4}\dot{5}$ (the recurring decimal).
Multiply by 100: $$100x = 45.\dot{4}\dot{5}$$
Subtract the two equations. We do this to remove the recurring part: $$
\begin{aligned}
100x – x &= 45.\dot{4}\dot{5} – 0.\dot{4}\dot{5} \\
99x &= 45
\end{aligned}
$$Solve for $x$:
$$x = \tfrac{45}{99} = \tfrac{5}{11}$$
✅ So, $0.454545\ldots = \tfrac{5}{11}$.
Converting Repeating Decimals to Fractions (Example) (1:17)
Sometimes the recurring part of a decimal does not start immediately after the decimal point.
For example:
$$0.1666\ldots$$
has the digit 1 first, and then the 6 repeats forever.
We can still turn this recurring decimal into a fraction. Here’s how:
🔮 Example: Writing 0.1666… in a fraction:
Let $x = 0.1\dot{6}$ (the recurring decimal).
Multiply by 10: $$10x = 1.\dot{6}$$
Multiply again by 10 (so that the recurring part lines up): $$100x = 16.\dot{6}$$
:Subtract the two equations. We do this to remove the recurring part: $$
\begin{aligned}
100x – 10x &= 16.\dot{6} – 1.\dot{6} \\
90x &= 15
\end{aligned}
$$Solve for $x$:
$$x = \tfrac{15}{90} = \tfrac{1}{6}$$
✅ So, $0.1666\ldots = \tfrac{1}{6}$.
📂 Flashcards: Converting Repeating Decimals to Fractions Method and Examples
🍪 Quiz: Practice Converting Recurring Decimals to Fractions
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